03/10/09 @ Juelich
－ a selected overview
Ming-Che Chang
Department of Physics
National Taiwan Normal University
Qian Niu
Department of Physics
The University of Texas at Austin
1

Taiwan

Paper/year with the title “Berry phase” or
“geometric phase”
80
70
60
50
40
30
20
10
0
1990 1992 1994 1996 1998 2000 2002 2004 2006 2008
3

 Introduction (30-40 mins)
 Quantum adiabatic evolution and Berry phase
 Electromagnetic analogy
 Geometric analogy
 Berry phase in solid state physics
4

Fast variable and slow variable
H +
2 molecule
  i i
) e － electron; {nuclei} nuclei move thousands of times slower than the electron
Instead of solving time-dependent Schroedinger eq., one uses
Born-Oppenheimer approximation
• “Slow variables R i
” are treated as parameters λ ( t )
(Kinetic energies from P i are neglected)
• solve timein dependent Schroedinger eq.
H r p
  n ,
 x

E n ,

 n ,

“snapshot” solution

Adiabatic evolution of a quantum system
• Energy spectrum:
E (λ ( t )) x x n +1 n n -1
• After a cyclic evolution


( )
 
(0)
 n

T
 e
 i

0
T d t ' E n
( t ')  n
Dynamical phase
0 )
0 λ( t )
• Phases of the snapshot states at different λ’s are independent and can be arbitrarily assigned
 n
 t )
 e i n
)
 n

• Do we need to worry about this phase?
6

No!
• Fock, Z. Phys 1928
• Schiff, Quantum Mechanics (3rd ed.) p.290
Pf : Consider the n -th level,


 i
( ) e n e
 i
 t
0
' n
( ') 
H


( )
 i

 t

 n ,

 n
 i
 n ,


 
 n ,


≡
A n
( λ )
0
Redefine the phase,

' n ,

 e i n
A n
’ ( λ )

A n
( λ )
 n ,


  n
 
Choose a
 (λ) such that ,
A n
’ ( λ )=0
Thus removing the extra phase
Stationary, snapshot state
H
 n ,


E n
 n ,

7

One problem:

A
 does not always have a well-defined (global) solution
Vector flow A
Contour of

Vector flow A
Contour of

 is not defined here
C
C

C
A d
 
0

C
A d
 
0
8

M. Berry, 1984 :
Parameter-dependent phase NOT always removable!


 i

  T i dt
0


(0)

C
Index n neglected
• Berry phase
(path dependent)

C
 
 i

 


 d
 
0
Berry’s face
• Interference due to the Berry phase if

C
  
1 2

0, then
 
1

2

  
1 2


0
Phase difference
1
1 a b a
C
2 － 2
9

Some terminology
•
Berry connection (or Berry potential)
A
  i





 
3
•
Stokes theorem (3-dim here, can be higher)

C
 
C
A d
 
S

A da
•
Berry curvature (or Berry field)

1
F A
 i



 




S
C
•
Gauge transformation (Nonsingular gauge, of course)


A

F
 e i

C

A



F

C


 


Redefine the phases of the snapshot states
Berry curvature nd Berry phase not changed 10

2

Analogy with magnetic monopole
Berry potential (in parameter space) Vector potential (in real space)
A
  i






Berry field (in 3D)
F A

Berry phase

C

=

S

C
A
  d

Chern number
2
1


S
F
  da
 integer
Magnetic field
 
Magnetic flux
  
C
=

S
( )
 dr
Dirac monopole
4
1


S
B
  da
 integer
11

Example: spin-1/2 particle in slowly changing B field
• Real space z
• Parameter space
B z x S
C
H
 
B
 
B
B
 
Level crossing at B =0
E ( B )
 y
B

B x
B y
C
(a monopole at the origin)
Berry curvature

( )
 
B


, B
 
B


, B

1
2 B
2
Berry phase


=

S
 
1

2 spin × solid angle 12

Experimental realizations :
Bitter and Dubbers , PRL 1987 Tomita and Chiao, PRL 1986
13

Geometry behind the Berry phase
Why Berry phase is often called geometric phase?
base space
Examples:
• Trivial fiber bundle
(= a product space)
R 1 x R 1 fiber R 1
R 1 base fiber space
Fiber bundle
• Nontrivial fiber bundle
Simplest example: Möbius band
15

Fiber bundle and quantum state evolution
(Wu and Yang, PRD 1975)
Fiber space: inner DOF , eg., U(1) phase

Base space: parameter space
• Berry phase = Vertical shift along fiber
(U(1) anholonomy)
• Chern number n n

2
1


S da F For fiber bundle
～ Euler characteristic χ
 
2
1


S da G For 2-dim closed surface
χ = 2 χ = 0
χ = － 2
16

 Introduction
 Berry phase in solid state physics

20

Berry phase in condensed matter physics, a partial list:
 1982 Quantized Hall conductance (Thouless et al)
 1983 Quantized charge transport (Thouless)
 1984 Anyon in fractional quantum Hall effect (Arovas et al)
 1989 Berry phase in one-dimensional lattice
(Zak)
 1990 Persistent spin current in one-dimensional ring (Loss et al)
 1992 Quantum tunneling in magnetic cluster (Loss et al)
 1993 Modern theory of electric polarization
(King-Smith et al)
 1996 Semiclassical dynamics in Bloch band (Chang et al)
 1998 Spin wave dynamics
(Niu et al)
 2001 Anomalous Hall effect (Taguchi et al)
 2003 Spin Hall effect (Murakami et al)
 2004 Optical Hall effect
(Onoda et al)
 2006 Orbital magnetization in solid (Xiao et al)
 … 21

Berry phase in condensed matter physics, a partial list:
 1982 Quantized Hall conductance (Thouless et al)
 1983 Quantized charge transport (Thouless)
 1984 Anyon in fractional quantum Hall effect (Arovas et al)
 1989 Berry phase in one-dimensional lattice
(Zak)
 1990 Persistent spin current in one-dimensional ring (Loss et al)
 1992 Quantum tunneling in magnetic cluster (Loss et al)
 1993 Modern theory of electric polarization
(King-Smith et al)
 1996 Semiclassical dynamics in Bloch band (Chang et al)
 1998 Spin wave dynamics
(Niu et al)
 2001 Anomalous Hall effect (Taguchi et al)
 2003 Spin Hall effect (Murakami et al)
 2004 Optical Hall effect
(Onoda et al)
 2006 Orbital magnetization in solid (Xiao et al)
 …

Berry phase in solid state physics
 Persistent spin current
 Quantum tunneling in a magnetic cluster
 Modern theory of electric polarization
 Semiclassical electron dynamics
 Quantum Hall effect (QHE)
 Anomalous Hall effect (AHE)
 Spin Hall effect (SHE)
• Persistent spin current
• Quantum tunneling
Spin
• AHE
• SHE
Bloch state
• Electric polarization
• QHE
23

Electric polarization of a periodic solid
P

1
V

3 d r r

( )
 well defined only for finite system
(sensitive to boundary)
 or, for crystal with well-localized dipoles
(Claussius-Mossotti theory)
• P is not well defined in, e.g., covalent crystal:
P
Choice 1 …
Unit cell
+ －
…
P
Choice 2 … － + …
• However, the change of P is well-defined
Δ P
…
Experimentally, it’s Δ P that’s measured
…
33

Modern theory of polarization
One-dimensional lattice (λ=atomic displacement in a unit cell)
P
 q
L
 nk
  nk r
  nk
I ℓℓ -defined
  nk
( r )
 e
 u nk
Resta, Ferroelectrics 1992
However, dP / dλ is well-defined, even for an infinite system !
 dP d
 d
 
P (

2
)

P

1
King-Smith and
Vanderbilt, PRB 1993 where P
 q

 n q
  n
BZ

2
 n dk
2
 u
 nk 
 i u k
 n k
Berry potential
• For a one-dimensional lattice with inversion symmetry
( if the origin is a symmetric point)
 n

0 or

(Zak, PRL 1989)
• Other values are possible without inversion symmetry 34

Berry phase and electric polarization
… g
1
=5 g
2
=4
Dirac comb model
…
0 b a
Rave and Kerr,
EPJ B 2005
 
1 q
 
2

1
Lowest energy band:
γ
1
← g
2
=0
γ
1
=π r =b/a similar formulation in 3-dim using Kohn-Sham orbitals
35
Review: Resta, J. Phys.: Condens. Matter 12, R107 (2000)

Berry phase in solid state physics
 Persistent spin current
 Quantum tunneling in a magnetic cluster
 Modern theory of electric polarization
 Semiclassical electron dynamics
 Quantum Hall effect
 Anomalous Hall effect
 Spin Hall effect
36

Semiclassical dynamics in solid
Limits of validity: one band approximation
Negligible inter-band transition.
“never close to being violated in a metal”
E ( k ) x x n +1 n dr dt dk
  eE
 er

B dt

1

E n
 k
0
• Lattice effect hidden in
E n
( k )
• Derivation is harder than expected
Explains (Ashcroft and Mermin, Chap 12)
• Bloch oscillation in a DC electric field, quantization → Wannier-Stark ladders
• cyclotron motion in a magnetic field, quantization → LLs, de Haas - van Alphen effect
…
2π
37 n -1

Semiclassical dynamics - wavepacket approach r W
1. Construct a wavepacket that is localized in both r -space and k -space
(parameterized by its c.m.) c k W c
2. Using the time-dependent variational principle to get the effective Lagrangian for the c.m. variables
L eff
( , c
; , c
)

W i

 t

H W
3. Minimize the action S eff
[ r c
( t ), k c
( t )] and determine the trajectory ( r c
( t ), k c
( t ))
→ Euler-Lagrange equations
Wavepacket in Bloch band:
L eff

 k c


  c k c

R n

E r k n c c
)
Berry potential
(Chang and Niu, PRL 1995, PRB 1996)
38

Semiclassical dynamics with Berry curvature
Simple and
Unified dk
  eE
 er

B dt dr dt

1

E n
 k
 k
  n
( k )
“Anomalous” velocity Cell-periodic
Bloch state
Berry curvature
 n
( ) i k u nk
  k u nk
Wavepacket energy
( , n c c
)

( ) n c
 e
2 m
L ( k )

B n c
Bloch energy
Zeeman energy due to spinning wavepacket
L k
   r c
  v W
If B =0, then d k /d t // electric field
→ Anomalous velocity ⊥ electric field
•
(integer) Quantum Hall effect
•
(intrinsic) Anomalous Hall effect
•
(intrinsic) Spin Hall effect

 Why the anomalous velocity is not found earlier?
In fact, it had been found by
• Adams, Blount, in the 50’s
 Why it seems OK not to be aware of it?
For scalar Bloch state ( non -degenerate band):
•
Space inversion symmetry
    n
( k ) n both symmetries
•
Time reversal symmetry
    n
( k ) n
 n
( )

0,
 k
 When do we expect to see it?
• SI symmetry is broken
•
TR symmetry is broken
•
spinor Bloch state (degenerate band)
Also, • band crossing
 n
( )

0
← electric polarization
← QHE
← SHE
← monopole
40

Berry phase in solid state physics
 Persistent spin current
 Quantum tunneling in a magnetic cluster
 Modern theory of electric polarization
 Semiclassical electron dynamics
 Quantum Hall effect
 Anomalous Hall effect
 Spin Hall effect
41

Quantum Hall effect
(von Klitzing, PRL 1980)
2 DEG classical
3 quantum
2
1
Increasing B 1/B
CB
AlGaAs z
GaAs
E
F
E
F
Each LL contributes one e 2 /h
Increasing B
B=0
LLs
2 DEG
VB
Density of states
42

Semiclassical formulation
Equations of motion
(In one Landau subband) dr dt dk
  eE dt

1

E
 k
 k
 
Magnetic field effect is hidden here
J
  e
 filled
2 r
=0
 e
2
E
  filled
2 d k
(2

)
2

( k )

J x
=
 
 e
2 h
1
2

 fille d
2 d k
 z


E y
Hall conductance

H
Quantization of Hall conductance
(Thouless et al 1982)
1
2


B Z d
2 k
 z
( k )
 intege r n
 
H
 n e
2 h
Remains quantized even with disorder, e-e interaction
(Niu, Thouless, Wu, PRB, 1985)
43

Quantization of Hall conductance (II)
Brillouin zone
For a filled Landau subband

BZ d k
 z
( )
 

BZ
Counts the amount of vorticity in the BZ due to zeros of Bloch state
(Kohmoto, Ann. Phys, 1985)
In the language of differential geometry, this n is the (first) Chern number that characterizes the topology of a fiber bundle
(base space: BZ; fiber space: U(1) phase)
44

Berry curvature and Hofstadter spectrum
2DEG in a square lattice + a perpendicular B field tight-binding model:
(Hofstadter, PRB 1976)
Landau subband
LLs


0

1
3
Magnetic flux (in Φ
0
) / plaquette 45

Bloch energy E(k) Berry curvature Ω(k)
C
1
= 1
C
2
=

2
C
3
= 1
46

Re-quantization of semiclassical theory
L eff

 k

 k R
Bohr-Sommerfeld quantization
1
2
C m

 k
 dk

 dz
  m
Berry phase

( C m
)

R dk
C m


( C m
)
2 2
 eB
Would shift quantized cyclotron energies (LLs)
• Bloch oscillation in a DC electric field, re-quantization
→
Wannier-Stark ladders
• cyclotron motion in a magnetic field, re-quantization
→
LLs, dHvA effect
• …
Now with
Berry phase effect!
48

cyclotron orbits (LLs) in graphene ↔ QHE in graphene
E
Dirac cone
B
ρ
L
σ
H
Cyclotron orbits k

( )
   
C
 
 
F k
 k
2 
2


 m

2 2
C


 eB

E n
 v
F
2 eB

 n

2 2
C



Novoselov et al, Nature 2005
49

Berry phase in solid state physics
 Persistent spin current
 Quantum tunneling in a magnetic cluster
 Modern theory of electric polarization
 Semiclassical electron dynamics
 Quantum Hall effect
 Anomalous Hall effect
 Spin Hall effect
Mokrousov’s talks this Friday
Buhmann’s next Thu (on
QSHE)
Poor men’s, and women’s, version of QHE, AHE, and SHE
50

Anomalous Hall effect
(Edwin Hall, 1881):
Hall effect in ferromagnetic (FM) materials
FM material
ρ
H saturation slope=R
N
The usual Lorentz force term

H

R
N
H
 
AH
( H ) ,
Anomalous term

AH
( H )

R
A H
M ( H )
R
AH
M
S
H
Ingredients required for a successful theory:
• magnetization (majority spin)
• spin-orbit coupling
(to couple the majority-spin direction to transverse orbital direction)
51

Intrinsic mechanism (ideal lattice without impurity)
• Linear response
• Spin-orbit coupling
• magnetization gives correct order of magnitude of
ρ
H for Fe, also explains that’s observed in some data

AH

L
2
52

Alternative scenario:
Extrinsic mechanisms
(with impurities)
Smit, 1955: KL mechanism should be annihilated by
(an extra effect from) impurities
• Skew scattering (Smit, Physica 1955)
～ Mott scattering
Spinless impurity e －

AH

L
• Side jump (Berger, PRB 1970) anomalous velocity due to electric field of impurity ～ anomalous velocity in KL
 
AH L
2
 
1 A
(Crépieux and Bruno,
PRB 2001) e －
2 (or 3) mechanisms:

AH

( )

L

( )

L
2
In reality, it’s not so clear-cut !
53
Review: Sinitsyn, J. Phys: Condens. Matter 20, 023201 (2008)

CM Hurd, The Hall Effect in Metals and Alloys (1972)
“ The difference of opinion between Luttinger and
Smit seems never to have been entirely resolved.”
30 years later:
Crepieux and Bruno, PRB 2001
“ It is now accepted that two mechanisms are responsible for the AHE: the skew scattering… and the sidejump…”
54

However,
Science 2001
Science 2003
And many more …
Karplus-Luttinger mechanism:
Mired in controversy from the start, it simmered for a long time as an unsolved problem, but has now re-emerged as a topic with modern appeal. – Ong @ Princeton
55

Old wine in new bottle
Karplus-Luttinger theory (1954)
= Berry curvature theory (2001)

AH
 e
2
 filled
3 d k
(2

)
3

( k )

0
→ intrinsic AHE
• same as Kubo-formula result
• ab initio calculation
Berry curvature of fcc Fe
(Yao et al, PRL 2004)
Ideal lattice without impurity
56

• classical Hall effect
+++++++
B
 Lorentz force
• anomalous Hall effect
B
↑↑↑↑↑↑↑
↑ ↑ ↑ ↑
 Berry curvature
 Skew scattering
0
• spin Hall effect
↑↑↑↑↑↑↑
↑↑↑↑
↑↑↑↑
↑↑↑↑↑↑↑
No magnetic field required !
 Berry curvature
 Skew scattering
E
F
↑
0 charge
E
F
0
E
F
↑
↑↓
L y charge spin
↓ spin
↓
L y
L y
57

Murakami, Nagaosa, and Zhang, Science 2003:
Intrinsic spin Hall effect in semiconductor
Band structure
• Spin-degenerate Bloch state due to Kramer’s degeneracy
→ Berry curvature becomes a
2x2 matrix (non-Abelian)
•
(from Luttinger model)
Berry curvature for HH/LH
Ω
HH
 
3
2 k
ˆ k
2
σ z
4-band
Luttinger model
The crystal has both space inversion symmetry and time reversal symmetry !
dx

 dt n

( )
 e k
E
 Ω n
Spin-dependent transverse velocity
→ SHE for holes
58

Only the HH/LH can have SHE?
8-band
Kane model
: Not really
• Berry curvature for conduction electron:
Ω   σ 
O k
1
( ) spin-orbit coupling strength
Dirac’s theory mc 2 electron positron
Chang and Niu, J Phys, Cond Mat 2008
•
Berry curvature for
free electron (!):
Ω  

C
2
2

C
σ


O k
1
( )
/ mc
 
12
10 m
59

Observations of SHE (extrinsic)
Science 2004
Nature 2006
Observation of Intrinsic SHE?
Nature Material 2008
60

• Summary
Spin
• Persistent spin current
• Quantum tunneling
• AHE
• SHE
Bloch state
• Electric polarization
• QHE
• Three fundamental quantities in any crystalline solid
E ( k )
Bloch energy
Berry curvature
Ω( k ) L ( k )
Orbital moment
(Not in this talk)
61

Slides : http://phy.ntnu.edu.tw/~changmc/Paper
Reviews: • Chang and Niu, J Phys Cond Matt 20, 193202 (2008)
• Xiao, Chang, and Niu, to be published (RMP?)
63